Question: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-5x-7y &= 1 \\ 6x+2y &= 6\end{align*}$
Answer: Begin by moving the $y$ -term in the second equation to the right side of the equation. $6x = -2y+6$ Divide both sides by $6$ to isolate $x$ $x = {-\dfrac{1}{3}y + 1}$ Substitute this expression for $x$ in the first equation. $-5({-\dfrac{1}{3}y + 1}) - 7y = 1$ $\dfrac{5}{3}y - 5 - 7y = 1$ Simplify by combining terms, then solve for $y$ $-\dfrac{16}{3}y - 5 = 1$ $-\dfrac{16}{3}y = 6$ $y = -\dfrac{9}{8}$ Substitute $-\dfrac{9}{8}$ for $y$ in the top equation. $-5x-7( -\dfrac{9}{8}) = 1$ $-5x+\dfrac{63}{8} = 1$ $-5x = -\dfrac{55}{8}$ $x = \dfrac{11}{8}$ The solution is $\enspace x = \dfrac{11}{8}, \enspace y = -\dfrac{9}{8}$.